Phase operators, temporally stable phase states, mutually unbiased bases and exactly solvable quantum systems. (English) Zbl 1186.81052

Summary: We introduce a one-parameter-generalized oscillator algebra \(\mathcal A_k\) (that covers the case of the harmonic oscillator algebra) and discuss its finite- and infinite-dimensional representations according to the sign of the parameter \(\kappa \). We define an (Hamiltonian) operator associated with \(\mathcal A_k\) and examine the degeneracies of its spectrum. For the finite (when \(\kappa < 0\)) and the infinite (when \(\kappa \geq 0\)) representations of \(\mathcal A_k\), we construct the associated phase operators and build temporally stable phase states as eigenstates of the phase operators. To overcome the difficulties related to the phase operator in the infinite-dimensional case and to avoid the degeneracy problem for the finite-dimensional case, we introduce a truncation procedure which generalizes the one used by Pegg and Barnett for the harmonic oscillator. This yields a truncated-generalized oscillator algebra \(\mathcal A_{\kappa ,s}\), where \(s\) denotes the truncation order. We construct two types of temporally stable states for \( \mathcal A_{\kappa ,s}\) (as eigenstates of a phase operator and as eigenstates of a polynomial in the generators of \(\mathcal A_{\kappa ,s})\). Two applications are considered in this paper. The first concerns physical realizations of \(\mathcal A_{\kappa}\) and \(\mathcal A_{\kappa ,s}\) in the context of one-dimensional quantum systems with finite (Morse system) or infinite (Pöschl-Teller system) discrete spectra. The second deals with mutually unbiased bases used in quantum information.


81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
11C08 Polynomials in number theory
81R15 Operator algebra methods applied to problems in quantum theory
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
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